I could be wrong, but I'm pretty sure Basecamp is not at all saying people should not reveal, let alone actively hide, their sexual orientation, gender identity, or anything like that.

I think anyone else at the company who considered such a thing political and tried to tell that person or the chat that what they're doing is political would be the only individual who's violating the "no politics at work" policy (perhaps in addition to other policies, too).

I understand your concern in general, here, but I don't see any reason why "gays are gonna get told to stop mentioning their husbands like we used to" is a thing that would start happening as a result of this policy change at Basecamp. I very strongly share your opinion that a conservative person who would do such a thing is deeply in the wrong, but I don't really think I see how it would apply to this particular situation.

I don't think "banning political discussion" at all carries the implications you're concerned about at most companies, and I think if it did happen at such a company it would be the thing that's violating the policy.